On Noether’s Theorem and the Various Integrals of the Damped Linear Oscillator

Abstract

Noether’s theorem provides deep insight into the connection between analytical mechanics and the integrals of dynamic systems, specifically, showing how symmetries of the action integral are connected to the integrals of motion. To demonstrate Noether’s theorem, the harmonic oscillator is often used as a simple example problem. Presentations in the literature, however, often focus on the single absolutely-invariant symmetry for this problem. This paper presents a complete application of Noether’s theorem to the damped harmonic oscillator, including general solutions of the divergence-invariant Killing equations and the associated integrals for all underdamped, critically-damped, and overdamped cases. This treatment brings forward several interesting issues. Five different symmetries produce independent solutions to the Killing equations, but of course, only two independent integrals exist for this second-order system. Also, integrals of a particular desired form may not be produced directly from Noether’s theorem and are referred to as non-Noether or asymmetric integrals. For the damped oscillator, one such example is the time-independent integrals, referred to as motion constants.